Chapter 0, Problem 18RE

To determine

To check: the graph of the symmetric about the $y$ -axis, the origin, or neither.

Answer to Problem 18RE

The given function is symmetric about the $y$ -axis.

Explanation of Solution

Given:

  $y=e^{-x^2}$

Concept used:

A function $f\left(x\right)$ is even if $f\left(-x\right)=f\left(x\right)$ and it is odd if $f\left(-x\right)=-f\left(x\right)$ .

Even functions are symmetric about the $y$ -axis, and the odd functions are symmetric about the origin.

Calculation:

Let $f\left(x\right)=e^{-x^2}$ .

Now,

  $\begin{array}{c}f\left(-x\right)=e^{-{\left(-x\right)}^2}\\ =e^{-x^2}\\ =f\left(x\right)\end{array}$

Therefore the given function is even.

Hence the given function is symmetric about the $y$ -axis.